13th International Conference on Fracture June 16–21, 2013, Beijing, China -1- Anti-plane problem of a lip-shape crack in one-dimensional hexagonal quasi-crystal materials Jing Yu1,2, JunHong Guo1,* , YongMing Xing1 1 College of Science, Inner Mongolia University of Technology, Hohhot 010051, China 2 College of general education, Inner Mongolia Normal University, Hohhot 011517, China * Corresponding author: jhguo@imut.edu.cn Abstract By introducing a conformal mapping and using the complex variable function method, the fracture behavior of a lip-shape crack in one-dimensional hexagonal quasi-crystals materials is investigated under anti-plane loading at infinity. The expressions for stress, strains, displacements and field intensity factors of the phonon and the phason fields in the vicinity of the crack tip are obtained. When the height of the lip-shape crack approaches to zero, the present results can be reduced to the solutions of the Griffith crack Keywords one-dimensional hexagonal quasi-crystals; lip-shape crack; complex variable function method; stress intensity factor 1. Introduction The discovery of quasi-crystals (QCs) in 1984 is a significant breakthrough for condensed matter physics, which won the Nobel’s award in 2011 [1]. A theoretical description of the deformed state of QCs requires a combined consideration of interrelated phonon and phason fields. The phonon field describes the motion of lattices in physical space, while the phason field describes quasiperiodic arrangement of atoms in the complementary orthogonal space, which interact with one another. Since the discovery of QCs, they have attracted the extensive attention of researchers engaged in experimental and theoretical work. A quantity of significant achievements of QCs have been done [2-11] recent years. Experiments have shown that quasi-crystals are quite brittle and the defects of quasicrystalline materials have been observed [12,13]. When quasicrystalline materials are subjected to mechanical stresses in service, the propagation of flaws or defects produced during their manufacturing process may result in premature failure of these materials. Therefore, the study of crack problem of quasicrystalline materials is meaningful both in theoretical and practical applications. At present, the study on the fracture problems of quasicrystalline materials is mainly confined to relatively simple defects. Thus, the elastic problem of one-dimensional (1D) hexagonal quasicrystal materials becomes the primary object and made many of significant achievements. A moving screw dislocation in 1D hexagonal QCs was investigated [14]. The exact solutions of a semi-infinite crack and two semi-infinite collinear cracks in a strip of 1D hexagonal QCs were obtained [15,16]. The interaction of between dislocations and cracks in 1D hexagonal QCs were considered by the complex variable function method. Very recently, the analytical solutions of several complicated defects such as cracks originating from holes in 1D hexagonal QCs were obtained [17-19]. In this paper, by using the Stroh-type formulism for anti-plane deformation in 1D hexagonal QCs, the fracture mechanic of a lip-shape crack in a 1D hexagonal QC is investigated under uniform remote anti-plane shear loadings of the phonon field and the phason field. By introducing a conformal mapping and using the complex variable function method, which is further solved analytically. The expressions for stress, strains, displacements and field intensity factors of the
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